The prediction of mechanical performance of isotactic
polypropylene on the basis of processing conditions
Citation for published version (APA):
Caelers, H. J. M., Govaert, L. E., & Peters, G. W. M. (2016). The prediction of mechanical performance of
isotactic polypropylene on the basis of processing conditions. Polymer, 83, 116-128.
https://doi.org/10.1016/j.polymer.2015.12.001
DOI:
10.1016/j.polymer.2015.12.001
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The prediction of mechanical performance of isotactic polypropylene
on the basis of processing conditions
H.J.M. Caelers, L.E. Govaert, G.W.M. Peters
*Department of Mechanical Engineering, Materials Technology Institute, Eindhoven University of Technology, P.O.Box 513, 5600 MB, Eindhoven, The Netherlands
a r t i c l e i n f o
Article history:
Received 30 September 2015 Received in revised form 30 November 2015 Accepted 1 December 2015 Available online 18 December 2015 Keywords: Semi-crystalline polymer Structure-property relations Lamellar thickness Yield stress
a b s t r a c t
A strategy is presented to predict the yield kinetics following from different thermomechanical histories experienced during processing in non-isothermal quiescent conditions. This strategy deals with three main parts, i.e. processing, structure and properties. In thefirst part the applied cooling conditions are combined with the crystallization kinetics and the cooling history of the material is calculated. From this history the lamellar thickness distributions are predicted in the second part. Finally, in the third part these distributions are used to predict yield stresses. Experimental validation is carried out for all the different parts of the strategy. In situ temperature measurements, lamellar thickness distributions from SAXS experiments and yield stresses measured in uniaxial tensile deformation are performed for vali-dation purposes. The versatility is investigated by applying this procedure on two different iPP grades. The yield stress predictions show good agreement with the experimentally obtained results in two separate deformation mechanisms, and only a few parameters are dependent on the specific iPP grades that were used here. Moreover, it is shown that the average lamellar thickness is sufficient to predict the yield stress, and that the width of lamellar thickness distributions does not have to be taken into account. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction
Polymers are used in a wide spectrum of applications ranging from packaging to structural engineering. Polyolefins, specifically polyethylene and polypropylene, form a substantial part of the synthetic polymers used because of their low costs, ease of manufacturing and versatility. To illustrate, these materials are used in extrusion processes (pipes),film blowing processes (pack-aging) and injection molding processes (structural applications). Within this specific class of materials multiple variants of poly-propylene exist, e.g. isotactic-, syndiotactic-, atactic polypoly-propylene and many copolymers. Their properties are related to the chemical structure, in particular the presence of regularity[1], since it allows polypropylene (iPP and sPP) to partially crystallize upon cooling. Due to the ability to crystallize the solidification takes place at higher temperatures as compared to aPP, largely affecting the mechanical properties. Other important aspects dominating the morphology and thereby the mechanical properties, are the pro-cessing conditions. Flow and cooling conditions are known to
largely affect the morphology and therewith the yield kinetics and overall mechanical response[2,3]. Since changes of these process-ing conditions throughout a product may therefore result in strong spatial variations of mechanical performance [4], an undesired consequence is that weak spots are typically present. In this work a first attempt is made to relate the mechanical properties to the morphology resulting from well-defined processing conditions.
The solid crystalline parts, present in iPP, are connected by chains surpassing the amorphous regions [5,6]. Some general findings on the relation between the crystals and the mechanical properties follow from several studies performed in the past. First, the Young's modulus increases with the degree of crystallinity, whereas the impact performance and the toughness decrease[7,8]. Furthermore, the yield stress appears to be strongly correlated to lamellar thickness[9e14]. This relation was rationally based on the nucleation and propagation of screw dislocations[12]in the crys-talline lamellae and thus on the lamellar thickness.
Besides the variations in the thickness of the crystalline domains (lamellae), multiple crystallographic structures can be present. In iPP, monoclinic
a
, pseudo-hexagonalb
, orthorhombicg
and mesomorphic unit cell structures [15,16] can be formed with alternating amorphous and rigid amorphous regions in between* Corresponding author.
E-mail address:g.w.m.peters@tue.nl(G.W.M. Peters).
Contents lists available atScienceDirect
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j o u r n a l h o m e p a g e :w w w . e l s e v ie r . c o m / l o c a t e / p o l y m e r
http://dx.doi.org/10.1016/j.polymer.2015.12.001
[17], affecting the mechanical properties as well. The presence of these regions together with the polymorphism makes it compli-cated to reveal the relationship between mechanical properties observed on a macroscopic scale to morphologies present at a microscopic or even smaller nanometer scale[18].
The crystallization process is kinetically controlled and there-fore local thermo-mechanical conditions experienced by the poly-mer during processing can have pronounced effects on the lamellar morphology that forms, as well as on the polymorphism within the crystals[19e21,4]. Structure development under processing con-ditions has been subject of substantial research carried out in the past [22e24]. When we focus on conditions imposed during a compression molding process, i.e. moderate non-isothermal quiescent conditions, it is found that in the case of neat iPP typi-cally
a
crystals develop[25]. Depending on the applied cooling rate, the crystallinity as well as the number and size of spherulites varies. Furthermore, this unavoidably results in variations of the lateral size of the crystal sheet-like domains, but more important for the yield kinetics, differences in the thickness will appear. The lamellar thickness is determined by the undercooling during the crystalli-zation process[26]and, therefore, directly related to the cooling rate. Van Erp et al. [3] specified this effect of cooling rate to investigate the structure property relation for iPP. On the other hand, several studies have been devoted to the development of model frameworks capable of quantitatively predicting the pro-cessing dependent crystal structures as a result of propro-cessing[25,27e30].
The main aim of this work was to make a coupling between the processing-structure and structure-property relation in a predic-tive way. The strategy chosen to accomplish this goal is schemati-cally shown inFig. 1and is divided into three main blocks. Different processing histories are obtained in terms of variable cooling rates. In thefirst block the processing dependent crystallization kinetics are predicted as a function of time and temperature. The time-temperature history follows from the heat equation, which is used in combination with the crystallization model proposed by van Drongelen et al.[25]to account for latent heat release. Tem-perature and pressure dependent growth rate and nucleation density are the most important parameters governing the crystal-lization process, whereas the boundary conditions together with the thermal contact resistance determine the temperature
evolution. In the second block the obtained evolution of crystal volume as a function of temperature is used in combination with the Lauritzen-Hoffman equation [26] to determine the lamellar thickness distributions resulting from the different cooling rates. Also the dependency of the molecular properties of the iPP chain on the crystallization temperature and lamellar thickness is deter-mined. Finally, in block three, the lamellar thickness is used to get the yield kinetics by making use of an empirical relation reported by van Erp et al.[3].
In the present study we willfirst give more detailed background information on 1) the crystallization model and the simplifications that are used, 2) the coupling to a structural feature, in this case lamellar thickness and 3) the relation between the lamellar thick-ness and the yield kinetics. Subsequently these three distinct parts are coupled and used to predict yield stresses resulting from well defined thermo-mechanical histories. The validity of this approach is experimentally shown for two iPP grades.
2. Experimental 2.1. Materials
Two isotactic polypropylene homopolymer grades were used: iPP-1 (Borealis HD234CF) with a weight averaged molar weight Mw¼ 310 kg/mol and a polydispersity Mw/Mn ¼ 3.4, and iPP-2 (Borealis HD601CF) with Mw ¼ 365 kg/mol and Mw/Mn ¼ 5.4. These two materials were chosen because they were used in several other crystallization studies in our group[25,31].
2.2. Sample preparation
To obtain samples with different thermal histories, sheet ma-terial with a thickness of 1 mm was compression molded from both the iPP-1 an iPP-2 grade. A mold, sample surface area of 100 cm2, was sandwiched in between stainless steel sheets (0.5 mm) and placed in a hot press, seeFig. 2. The stack was subsequently heated to 230 C and a force of 100 kN was applied stepwise. The sheets were kept under these conditions for 3 min to erase previous thermo-mechanical history. The solidification was induced by putting the stack in a cold press for 3 min, at temperatures varying from 20C to 90C (steps of 10C). To monitor the temperature
during solidification, a small calibrated thermocouple was embedded in the polymer. A fast acquisition datalogger (National Instruments Hi-speed USB 9162, sampling frequency 10 Hz) was used to record the temperature in-situ.
2.3. X-ray
Small angle X-ray scattering (SAXS) and wide angle X-ray diffraction (WAXD) experiments were performed at the Dutch-Belgian (DUBBLE) beamline BM26[32]of the European Synchro-tron and Radiation Facility (Grenoble, France). Quasi-isothermal crystallization experiments were performed with a custom modi-fied JHT-350 Linkam stage equipped with a pneumatically actuated temperature jump-stage[33]. The cold stage was set at tempera-tures of 100, 110 and 120C respectively. To monitor the temper-ature and quasi-isothermal crystallization in time, a small thermocouple was embedded in the polymer. In-situ WAXD ans SAXS patterns were recorded with acquisition rates of 20 frames per second. A wavelength of
l
¼ 1.04Å was used. The 2D SAXS patterns were recorded with a Pilatus 1 M detector and the WAXD patterns with a Pilatus 300 K detector, both with pixel size of 172 172m
m2, placed at approximately 1.42 and 0.30 m respectively.Single shots were obtained ex-situ from the compression mol-ded samples with the different cooling histories, with an acquisi-tion time of 10 s. All the acquired images were corrected for beam intensity and scattering of the empty sample cell.
3. WAXD
The obtained intensity profiles were plotted versus the scat-tering angle 2
q
. The weight fraction of the crystallinityc
w was determined with Eq.(1):cw¼Ctot Ca
Ctot (1)
where Ctot is the total scattered intensity and Cais the scattered intensity of an amorphous halo. The amorphous halo is measured on a quenched low tacticity polypropylene sample with negligible crystallinity, and scaled with the minimum between the (110)aand
(040)adiffraction peaks[34]. The volume fraction of the
crystal-linity is given by Ref.[9]:
c¼ cw rc cw rcþ 1cw ra (2)
where
r
aandr
care the density of the amorphous and crystalline phase respectively, taken from Ref.[35].4. SAXS
The scattered intensity was obtained as a function of the
scattering vector q which is given by:
q¼4p
l sinðqÞ (3)
where
q
is half of the scattering angle. In case of an isotropic system with a randomly oriented lamellar morphology the measured scattering intensity can be transposed into the 1D scattering in-tensity using Lorentz correction:I1ðqÞ ¼ IðqÞq2 (4)
Once this correction is performed and the electron density dif-ferences in one direction are known, the average lamellar thickness lccan be obtained from:
lc¼q2p I1;max
c (5)
where qI1;maxis the value for the magnitude of the scattering vector
q, corresponding to the maximum of the Lorentz corrected scat-tering intensity, and
c
is the crystalline volume fraction. This method provides an average lamellar thickness. Information about lamellar thickness distributions was obtained using the interface distribution function (IDF) which is the second derivative of the 1D-correlation functiong
1(r) (Eq.(7))[36,37], and is defined as:g1ðrÞ ¼d2gðrÞ dr2 ¼ 1 Q Zq∞ q0 I1ðqÞq2cosðqrÞ dq (6) with g1ðrÞ ¼1 Q Zq∞ q0 I1ðqÞcosðqrÞ dq (7)
where Q is the invariant and r is the real space. The interface dis-tribution function g1(r) can also be obtained by taking the inverse Fourier transform of the interference function G1(q)[38]
g1ðrÞ ¼ Zq∞ q0 G1ðqÞcosðqrÞ dq (8) in which G1ðqÞ ¼ limq/∞I1ðqÞq2 I1ðqÞq2: (9)
Since the Fourier transform requires integration from q¼ 0 to infinite, the experimentally accessable q range has to be extrapo-lated. The triangle rule is used to extrapolate to zero q, whereas the Porod law is used to extrapolate to infinite q. In an ideal two-phase system with sharp boundaries the Porod law predicts a decay in scattered intensity proportional to q4at large angles. In reality the intensity often deviates from such an ideal system because of electron density fluctuations and finite interfaces between the crystalline and the amorphous layers. When taking these de-viations into account, the adapted Porod law is given by Ref.[39]:
lim q/∞IobsðqÞ ¼ IbðqÞ þ Kp q4exp s2q2 (10)
where
s
is related to the interface thickness, Kpis the Porod con-stant and Ibthe scattering resulting from electron density fluctua-tions. The determination of the parameters required to correct fornon-ideality is a sensitive process which can be rather difficult in case of a noisy signal. For this reason an approach proposed by Hsiao et al.[40] is used, where constraints are used tofind the parameters required for the intensity corrections. Thisfit is based on two properties that have to be fulfilled by the interference function. First, the difference between the asymptote at large values of q, following from the Porod law, and the ideal scattered intensity should become zero. Moreover, as a second constraint, the interface distribution function should start from the origin and as a result the total area of G1(q) versus q should be zero. Minimization of Eq.(11), which is only valid in the porod region, and Eq.(12)gives the values for the Porod constant, the interface thickness and the liquid-like scattering.
lim
q/∞G1ðqÞ ¼ limq/∞
h
Kp ½IobsðqÞ IbðqÞq4exp
s2q2i¼ 0 (11) Zq∞ q0 G1ðqÞ dq ¼ Zq∞ q0 h
Kp ðIobsðqÞ IbðqÞÞq4exp
s2q2idq¼ 0
(12)
The background intensity following from the electron density fluctuation or liquid like scattering is expressed by
IbðqÞ ¼ a þ bq2þ cq4þ dq6 (13)
The most reliable average value and the distribution of the lamellar thickness, the long spacing and the amorphous regions can be obtained by deconvolution of the interface distance distribution function g1(r). Long spacings obtained by Bragg's law are typically considerably larger then the true values, especially when distri-butions are broad [41]. This will irrefutable result in erroneous lamellar thicknesses found with the method of combining WAXD and SAXS measurements Eq.(5).
4.1. Mechanical testing
A punch was used to cut typical dog-bone shaped tensile test samples (according to ASTM D1708) from the different compres-sion molded polymer sheets. A Zwick Z010 universal tensile tester equipped with a 2.5 kN load cell and a thermostatically controlled oven was used to perform the tensile tests at strain rates of 103s1. The tests were performed at 23C and 80C. In advance of the measurements at elevated temperatures, the tensile specimen was kept at the test temperature for 5 min (which is sufficient to achieve thermal equilibrium) before a pre-load of 0.2 MPa was applied. All tests were carried out at least in duplicate. Tensile tests were per-formed immediately after sample preparation to avoid effects of aging at room temperature[42e44].
5. Background 5.1. Thermal analysis
Cooling rate affects the crystallization process of iPP, but conversely, the cooling rate is influenced by the crystallization process because latent heat releases. This mutual influence of the cooling and crystallization process is elaborated in terms of a 1D conduction problem. In al the experimental test cases performed in this work the polymer layer is positioned in between layers of stainless steel. These are included in the model in order to get the boundary conditions right.
5.1.1. The heat balance
To predict the temperature profile in the polymer, the 1D heat equation for conduction is used:
rðP; x; TÞ$Cpðx; TÞ$vTvt ¼vxv lðx; TÞ$vT vx þ rðP; x; TÞ$DH$_x (14)
In this equation the specific heat Cp[J/kg K], the density
r
[kg/ m3] and the thermal conductivityl
[W/mK] are all functions of temperature and crystallinity. The effect of pressure on the density is not taken into account. The last term of Eq.(14)is the source term, representing the latent heat release due to crystallization[45]. The time derivative of the spacefilling, _x, follows directly from the crystallization model described in Section5.2and
D
H [J/kg] is the total enthalpy of transformation. To capture the phase depen-dent thermal properties a simple mixing rule, Eq. (15), is used which is similar for heat capacity, density and thermal conductivity.Cpðx; TÞ ¼ xCpscðTÞ þ ð1 xÞCpaðTÞ (15)
The heat capacity and the thermal conductivity are linearly proportional to the temperature, whereas the density is propor-tional to the reciprocal temperature[35]. Subscripts a and sc refer to the amorphous and semi-crystalline phase, respectively. Heat transfer in the aluminum and steel layers of the experimental setup is again described with the 1D heat equation. However, in that case the thermal properties are assumed to be constant and the source term disappears. Therefore Eq.(14)reduces to:
rCpvT vt¼ l v2T vx2 ! (16)
The parameters that were used in the heat equation are given in
Table 1.
5.1.2. Thermal contact resistance
The experimental setup consists of a stack of polymer-, and stainless steel layers. As a result of surface roughness or interstitial materials, a pressure dependent thermal contact resistance is present between the layers. Moreover, the state of the polymer, melt or solid, influences the thermal contact behavior.
This contact behavior is included in the model using Eq.(17). No data is available on pressure and state dependency, so the thermal contact resistance is assumed to be constant. It follows that the heat flux through the interface is given by:
4intðtÞ ¼
Tsurf 1 Tsurf 2
TCR (17)
where4int(t) [W/m2] is the heatflux from surface Tsurf1[K] to sur-face Tsurf2[K], and TCR [m2K/W] the thermal contact resistance. The ingoing heatflux (conduction) equals the flux through the inter-face, and the outgoing heatflux:
4inðtÞ ¼ 4intðtÞ ¼ 4outðtÞ (18)
This results in:
Table 1 List of constants.
r[kg/m3] Cp [J/kg K] l[W/mK]
lAdT dx left ¼ 4intðtÞ ¼ lB dT dx right (19)
were
l
Aandl
Bare the thermal conductivities that belong to ma-terial A and B respectively. This is approximated by using the temperature gradient over the neighboring grid points:lADT Dx left ¼Tsurf 1 Tsurf 2 TCR ¼ lB DT Dx right (20)
After discretization using afinite difference method, the heat equation is solved using an implicit Euler scheme. The thermal contact resistance is determined by using the initial slope of the cooling curve for the fastest cooling rate. The parameter values used in the model for the thermal contact resistance are given in
Table 2. For both iPP grades the same temperature dependent re-lations for density, specific heat and thermal conductivity are used
[35]which explains why thefitted thermal contact resistance be-tween the polymer and the stainless steel is different.
5.2. Crystallization kinetics
Crystallization of iPP is influenced by the chain architecture. Isotacticity is a key parameter with significant effects on crystal-linity, polymorphism [46] and crystallization temperature [47]. Molecular weight affects the crystallization temperature[48]. Be-sides these chain architectural features the thermomechanical history experienced during processing is of significant importance. In the absence offlow and shear, the arising morphology and the crystallographic structures present therein are determined by the cooling rate[49]and pressure[19]. In this work we focus on the relationship between processing and structure, which has been subject to substantial research, and is captured in multiple crys-tallization models, many of them lacking structural details. A vali-dated model to describe the temperature and pressure dependent crystallization behavior of iPP in detail, i.e. local nucleation density, spherulite size etc., was proposed by van Drongelen et al.[25]. This model framework is capable of predicting multiphase structure development in quiescent non-isothermal isobaric conditions. In this work a simplified version of this model is used which only allows monoclinic alpha phase formation. From WAXD measure-ments it is shown later that the model is applicable for the ther-momechanical histories assessed in this study.
5.2.1. The crystallization model
Crystallization is dominated by nucleation and growth. In quiescent conditions the nuclei grow radially until they finally impinge and reach complete spacefilling. This can be described with the Kolmogoroff equation[50], which gives the spacefilling as a result of nucleation and growth in an unconfined 3-dimensional space according to:
xðtÞ ¼cðtÞ
c∞ ¼ 1 expðf0ðtÞÞ (21)
where
c
(t) is the crystallized volume fraction at time t andc
∞is the crystallinity when equilibrium is reached. The expected crystallizedvolume fraction if no impingement would occur
f
0(t) is given by Ref.[50]: f0ðtÞ ¼4p 3 Zt ∞ dt0aðt0Þ 2 6 4 Zt ∞ duGðuÞ 3 7 5 3 (22)In this equation
a
(t)¼a
(T(t), p(t)) and G(u)¼ G(T(u), p(u)) are the (spherulitical) nucleation and growth rate respectively, both functions of temperature and pressure. In the special case of isothermal isobaric crystallization where growth rate and nucle-ation density (i.e. heterogeneous nuclenucle-ation) are constants, the spacefilling in timex
(t) reduces toxðtÞ ¼ 1 exp
4p3 NG3t3
(23)
which is known as the classical Avrami equation[51,52]. However, in this work non-isothermal crystallization is considered. There-fore, we start from the Kolmogoroff Eq. (22). To solve non-isothermal crystallization problems it is much easier to work with the Schneider rate equations which are basically a transform of this integral into a more suitable configuration. Now,
f
0(t) follows from the rate equations[53]:_f3¼ 8p _N ðf3¼ 8pNÞ
_f2¼ Gf3 ðf2¼ 8pRtotÞ
_f1¼ Gf2 ðf1¼ StotÞ
_f0¼ Gf1 ðf0¼ VtotÞ
(24)
where N is the number of nuclei (heterogeneous nucleation den-sity), _N is the nucleation rate, G is the spheruletic growth rate, Rtotis the sum of the spherulite radii, Stot is the total surface of the spherulites and their total volume is given by Vtot. These structural features can be obtained since the nucleation and growth calcu-lated via these equations, depend on the thermal history. The so-lution of these equations in isothermal conditions, and with a constant nucleation and growth rate, again results in Eq.(23). In this work the number of nuclei and the growth rate are tempera-ture and pressure dependent and described by the expressions (25) and (26) respectively, NðT; pÞ ¼ Nrefexp cn TðtÞ TNrefðpÞ (25) GðT; pÞ ¼ GmaxðpÞexp cg TðtÞ TGrefðpÞ 2 (26)
where Nref is the reference number of nuclei at the reference temperature TNref. Gmaxis the maximum growth rate at the refer-ence temperature TGref, p is the pressure and cnand cgare constants. The effect of pressure on the nucleation density in incorporated by a shift in the reference temperature, and for the growth rate a shift of the reference temperature and a change in the maximum growth rate parameter Gmax is included, according to the following equations,
Tk;ref ¼ Tk;ref0 þ zðp p0Þ$105 (27)
Gmax¼ G0maxexp
aðp p0Þ þ bðp p0Þ2
(28)
where T0
k;ref and G0max are the reference temperature and growth rate at atmospheric pressure p0in bar, and a, b and
z
are constants. The index k represents the growth (G) and nucleation (N).Table 2
List of parameter values.
TCR in [m2K/W] Press iPP-1 Press iPP-2
Stainless steele stainless steel 3$104 3$104
InFig. 3it is schematically shown how the effect of pressure shifts the nucleation density and the growth rate respectively.
Finally, when the nucleation density and the growth rate are adapted for the applied pressure and non-isothermal conditions, the spacefilling in time _x follows from:
_x ¼ ð1 xÞ _f0 (29)
An explicit Euler scheme is used to solve the crystallization model. The parameters required to describe the crystallization process are adopted from the work of van Drongelen et al.[25]. An overview is given inTable 3.
5.3. Deformation kinetics 5.3.1. Yield kinetics
To predict yield stresses resulting from well defined processing conditions wefirst look at the phenomena related to deformation kinetics. The behavior typically displayed by isotactic poly-propylene is shown inFig. 4(a). At low strains the stress increases linearly. With further increasing strain, the stress and the molecular mobility within the polymer increase as well. Ultimately, in the yield point, the molecular mobility is so high that the material deforms plastically at a rate equal to the applied strain rate. The stress associated with this point is defined as the maximum in the stress-strain response and called the yield stress. With increasing strain-rates higher molecular mobility is required for yielding. This is achieved by a higher stress level, explaining the rate dependency of the yield stress typically observed for polymers. Another way to induce mobility is raising the temperature. In the mechanical response this leads to decreasing yield stresses. After yielding strain softening takes place, leading to strain localization and subse-quently necking.
The yield kinetics, i.e. the yield stresses over a broad range of
temperatures and strain rates are shown in Fig. 4(b). From this figure it can directly be observed that the rate dependency at 23C is stronger then at 110 C At an intermediate temperature, for example at 80C, we can distinguish the two slopes for different ranges of deformation rateFig. 5(a). These different slopes originate from the fact that two separate deformation mechanisms are pre-sent, schematically represented inFig. 5(b). At high temperatures or low strain rates, only the process of crystal slip or intra-lamellar deformation determines the yield stress[54]. At lower tempera-tures, crystal slip or inter-lamellar deformation starts to actively contribute to the observed yield stresses[55].
Since the deformation processes act in parallel (stress additive), the observed kinetics can be described by taking the sum of the two separate processes. In this work this is done with the modified Ree-Eyring equation: stotal¼ X i¼I;II si¼ X i¼I;II kT Visinh1 _ε _ε0;iexpðDUi=kTÞ ! (30)
In this equation k is the Boltzman constant, T is the temperature in [K], _ε is the applied strain rate, V
i is the activation volume of deformation mechanism i,
D
Uiis the activation energy of mecha-nism i and_ε0;iis the rate constant. The temperature and strain rate are specified in the experimental section.5.3.2. The effect of processing
When we restrict ourselves to the influence of the cooling his-tory a decrease in crystallinity and lamellar thickness is found upon increasing cooling rates. The effect of these structural features on the mechanical properties and in particular the yield stress was
Fig. 3. a) Shift in the nucleation density as a result of pressure, b) Shift of the growth rate as a result of pressure. Adopted from Ref.[25].
Table 3
Model parameters.
Parameter iPP-1 iPP-2 Unit Nref 2.7$1014 1.2$1014 [m3] TNref 383 383 [K] cn 0.181 0.219 [K1] G0 max 4.5$106 4.81$106 [ms1] T0 Gref 363 363 [K] cg 2.3$103 2.3$103 [K2] a 1.60$109 1.60$109 [Pa1] b 0 0 [Pa2] z 0.0275 0.0275 [bar1]
Fig. 4. a) The stress strain response of iPP as a function of strain rate and temperature, and b) The yield kinetics of iPP.
investigated in Ref.[3], and it was found that the resistance against yield becomes stronger with lower cooling rates. Furthermore it was found that the activation volume and energy in the Ree-Eyring equation are independent of cooling rate, alpha nucleating agent or copolymer content. Moreover, the yield kinetics of multiple iPP grades including the ones used in this study, could be described perfectly with the same parameters. The only processing depen-dent variables in non-isothermal quiescent conditions were found to be the rate constants. Values for Viand
D
Uiare taken from van Erp et al.[3], and listed inTable 4.5.3.3. Relation between structural features and yield kinetics The only remaining Eyring parameters to be identified are the rate constants_ε0;i. The results of van Erp et al.[3]can straightfor-wardly be translated to obtain the relation between the logarithm of the rate constant and the lamellar thickness, shown inFig. 6.
The relations associated to the lines depicted inFig. 6are given by: log_ε0;I¼ 1:90llc c0þ 74:01 log_ε0;II¼ 0:76llc c0þ 28:12 (31)
with lc0¼ 1 nm. Making use of this empirical relation, which holds for multiple iPP-grades enables us to predict yield kinetics once the lamellar thickness is known. Although the amount of imperfections present within the crystalline domains is cooling rate dependent, a relation between lamellar thickness and rate constant is sufficient to describe the data measured by van Erp et al.[3]under the pro-cessing conditions applied in his work.
5.4. The relation between crystallization temperature and lamellar thickness
The lamellar thickness of crystals that grow at a certain tem-perature Tcis inversely proportional to the undercooling according to[26] lc¼ 2seT 0 m Dhf T0 m Tc þ dl; (32)
where
s
eis the surface free energy,D
hfis the enthalpy of fusion and Tcis the crystallization temperature. Tm0 is the equilibrium melting temperature, i.e. the melting temperature of a crystal with extremely large lamellar thickness. Here, the equilibrium crystal-lization temperature is not made pressure dependent since the pressures during the crystallization process are not far from at-mospheric pressure as a result of shrinkage due to crystal forma-tion. Thisfirst term in Eq.(32)represents a stable condition where the increasing surface energy 2s
eequals the reduction in free en-ergy obtained[56]. It should be emphasized that lcis the lamellar thickness prior to thickening. The last term in Eq.(32),d
l, is related to the tendency of the polymer to maximize the crystal growth and basically is a quantity arising from the kinetic nature of crystal growth, given by Ref.[26]:dl¼ kTc 2b0s " a0DhfDT þ 4sTc0 a0DhfDT þ 2sTc0 # (33)
where k is the Boltzman constant,
s
is the lateral surface free en-ergy, b0is the thickness of the surface layer and a0is the width ofFig. 5. a): Yield data measured on iPP-1 at a temperature of 80C. Two slopes corresponding to separate deformation mechanisms are indicated with I and II. b) Schematic
representation of the intra- and interlamellar deformation mechanism.
Table 4
List of parameter values.
Vi½nm3 DU i[kJ mol1]
Mechanism I 14.20 503.7 Mechanism II 4.44 158.0
Fig. 6. The relation between lamellar thickness and the rate constants, deduced from Ref.[3].
the molecule. At low and moderate cooling rates
d
may be approximated by Ref.[57]:dlykTc
b0s
(34)
The relation between lamellar thickness and crystallization temperature has been subject of many studies in the past, and an overview of some of these results is given inFig. 7. For example, Cheng et al. examined iPP with different degrees of stereo defects but similar molecular weight, and determined the lamellar thick-ness for samples crystallized isothermally at different temperatures
[47]. The conclusion that can be drawn from these results is that a unique relation between Tcand lcexists, independent of the degree of stereo defects. Iijima et al. used two iPP grades with similar isotacticity, but different molecular weights [58]. They found a relationship Tc versus lc that holds for both their isotactic poly-propylenes, independent of molecular weight. On the other hand, Lu et al.[48]used two isotactic polypropylenes with a much bigger difference in molecular weight. They found that as a result of increasing molecular weights, the surface free energy
s
eincreases, and thus a shift in Tcversus lc. This was interpreted as that for the lower molecular weight samples a relatively high amount of extended-chain crystallites are formed, whereas in case of higher molecular weight samples folded chain configurations are prefer-able. Experiments of Devoy et al. support this interpretation[59]. The T0m on the other hand was found to be unaffected, which is different from what Yamada et al.[60]found in their study. Lu et al. could reasonably resolve this latter disagreement by a crystalliza-tion theory proposed by Strobl[61]where the crystallization and melting are non-reversible processes. Based on thefindings pre-sented above the important conclusion is drawn that due to different molecular features present in specific iPP grades, de-viations in the relation between crystallization temperature and lamellar thickness are found. Therefore this relation is determined for the iPP grades used in this study. The assumption is made that the only variable parameter in Eq.(32)is the surface free energy
s
e. In agreement with Angelloz et al.[23]and Iijima et al. [58]the equilibrium melting temperature T0mis chosen at 193C, and used to describe the experimental data sets shown inFig. 7. An impor-tant note that should be emphasized is that the values found for lamellar thickness from X-ray experiments (partly) depend on the techniques used [41]. The interface distribution function as for
example used by Iijima et al. gives the most probable value for the lamellar thickness, whereas the correlation function as for example used by Cheng et al. gives the mean value. Finally the relation of Tc versus lccan be determined with DSC as well, as demonstrated by for example Wlochowicz et al.[62]. Another important note is that in case of extremely narrow molecular weight distributions trends will most likely be different, and only using
s
etofit the relation is insufficient. The data found for the iPP grades in this work will be described using the same set of parameters, adopted from Xu et al.[63], and are given inTable 5.
InFig. 7it can be seen that experimental data of several authors can be described using this set of parameters, and only varying the value of the surface free energy
s
e. InTable 6the values of this parameter are given.The relation between Tc and lc is determined for isothermal crystallization. For the non-isothermal experiments the temperature-time profiles are divided into discrete temperature steps and for each time step
D
t a crystal volume:DV ¼ Dt$_x (35)
is formed with an associated lamellar thickness obtained from Eq.
(32). The additional spacefilling _x, achieved during that specific time step follows from Eq.(29). As a result of the non-isothermal crystallization we predict lamellar thickness distributions. 6. Results and discussion
6.1. Temperature predictions
Crystallization from the melt begins with the formation of point-like nuclei that subsequently grow into spherulites. Due to the kinetics of the crystallization process different morphologies will arise when cooling rates are varied. In this study this was achieved by adjusting the temperature of the cold press, ranging from 20C to 90C. These temperatures act as boundary conditions in the thermal analysis. The melt was cooled from 220C before it was placed in the cold press. InFig. 8(a)a calculated cooling history of a 1 mm thick sheet in a 20C cold press is shown, as a function of time and position. The cooling rates are the highest close to the wall and the lowest in the center. The effect of the latent heat release can be recognised in the center since the decrease in temperature is followed by an increase. Subsequently the temperature decreases again.
To validate the predicted temperature profiles, in-situ time-temperature measurements were performed in case of the slowest and the fastest cooling rates assessed in this work. The small thermocouple with a thickness of approximately 0.4 mm was embedded in the polymer material to record the temperature. Since the thermocouple is relatively thick compared to the polymer sheet, an average temperature over the sheet thickness is measured and, therefore, a comparison is made with the calculated average time-temperature profile, shown inFig. 8(b). The position of the latent heat release contributions, featured by a plateau in the time-temperature profile, reveal that the crystallization temperatures
Fig. 7. The relation between lamellar thickness and crystallization temperature. Lines arefitted using an equilibrium melting temperature of 193C. Data are reproduced
from Refs.[47,58,48,62]. Table 5 Model parameters. Parameter Value se[J nm2] 146$1021 To m½K 466 Dhf[J nm3] 207$938$1024 b0[nm] 0.626 s[J nm2] 11.95$1021
decrease with increasing cooling rates. Moreover, it can be seen that the predictions are in good agreement with the experimental results, and that discrepancies arise mainly after the solidification. This can be explained by an increasing thermal contact resistance in the experiments, which is not included in the model. After solidi-fication the material shrinks and as a result the contact pressure reduces. Therefore, the predicted cooling rate is higher than the measured temperature decrease after crystallization.
6.2. X-ray analysis
It is assumed that, under the moderate cooling conditions applied here, only monoclinic alpha phase will be formed and that, therefore, the crystallization model could be simplified to the form presented in Section5.2. To justify this assumption the wide angle X-ray patterns measured on iPP-1 for all eight cooling rates are depicted inFig. 9(a). The crystallinities were al within 64± 5% and,
as expected, the characteristic
b
andg
peak are negligible with respect to thea
peak present at a scattering angle of 2q
y12.5o. Similar results are found for iPP-2.From Bragg's law the long spacing is obtained and via Eq.(5)this gives the lamellar thickness. The SAXS data of the isothermal ex-periments,Fig. 9(b), is also used to determine the interface distri-bution function. The corrections are explained in Section2.3, and a corrected 1D intensity pattern is visualised inFig. 10(a). Also the
interference function is depicted inFig. 10(a)and is used to calcu-late the interface distance distribution function using Eq.(8). A typical result is shown inFig. 10(b)where the IDF of iPP-1 measured at 120C is shown. Gaussians are used for deconvolution purposes. First, the most probable long spacing which is corresponding to the first minimum in g1(r), isfixed. Then, by using the crystallinity obtained via WAXD,± 5% the ratio between the most probable lamellar thickness and amorphous layer thickness is determined and Gaussians arefitted to obtain the thickness distributions of both the crystalline and amorphous domains. Thefirst maximum in g1(r) corresponds to
a
cross hatched structures and isfitted on the resulting part of the IDF[64]. The Gaussian distributions found via this deconvolution procedure are plotted inFig. 10(b)as well.An important observation is that, although the quasi-isothermal crystallization would result in nearly uniform lamellar thickness according to Eq. (32), wefind distributions. The full width half maximum (FWHM) of the Gaussians is on average 3.25 for the
Table 6
Surface free energy.
Author se[J nm2] Iijima 113.88$1021 Lu (Mn¼ 12 kg/mol) 131.40$1021 Cheng 135.78$1021 Wlochowitz 191.26$1021 Lu (Mn¼ 340 kg/mol) 219.00$1021
Fig. 8. An example of the predicted temperature profile as a function of time and position. iPP-1 with the cold press set at 20C (left) and the average time-temperature history of
iPP-1 samples prepared with different cold press temperatures.
isothermal crystallization experiments conducted on the two iPP-grades. Consequently, this lamellar thickness distribution is also included in the predictions. At every time step and corresponding crystallization temperature step, a lamellar thickness distribution is formed with this FWHM. In this work the lamellar thickness ob-tained from the IDF was used since it gives the most probable value for the lamellar thickness[41]. Additionally, the lamellar thickness obtained from the combination of SAXS and WAXD experiments Eq.
(5) provides an estimation of the error made using the latter approach. The results are shown inFig. 11and they are used in the following part by changing the parameters in Eq. (31)into the corrected ones given by:
log_ε0;I¼ 2:72lc;IDFl
c0 þ 78:88
log_ε0;II¼ 1:09lc;IDFl
c0 þ 30:09
(36)
6.3. Relation between Tc and lc
As explained in 3.4 experimental data of several authors could be described using Eq. (32) with an equilibrium crystallization temperature of 193C and a variable surface free energy. Although different experimental methods for lamellar thickness determina-tion yield differences in the values found, all presented data on the relation between crystallization temperature and lamellar thick-ness can be described accurately well using the parameters given in
Table 5and a specific value for
s
efor each data set. To obtain thes
e values for the two iPP-grades used in this study, quasi-isothermal cooling experiments were conducted. In-situ temperature mea-surements demonstrate that the crystallization took place at con-ditions close to isothermal, seeFig. 12(a). These temperatures were plotted as a function of the lamellar thickness obtained from the IDF.FromFig. 12(b)it can be seen that using a surface free energy which is higher for the iPP with the highest molecular weight (iPP-2,
s
e¼ 155.5,1021) and lower for the one with the low molecular weight (iPP-1,s
e ¼ 134.3$1021) gives good descriptions for the lamellar thickness as a function of the crystallization temperature. Although the value ofs
e is determined on a small number of experimental data points, it seems reasonable with respect to the data reported by other authors. Thisfits the expectations based on Lu et al.[48], eventhough the differences in molecular weight are so small that it is highly unlikely that this is the only molecular feature causing this difference. From the time-temperature history within the polymer sheet, combined with Eq.(32)and the FWHM of 3.25, the lamellar thickness can be calculated. Summation of the distri-butions obtained at the different increments during the non-isothermal crystallization process gives the lamellar thickness distribution as a function of the position within the polymer sheet. In Fig. 13(a) an example of such a calculated lamellar thickness distribution profile is shown. In the center where the cooling rate was the lowest, the formed lamellae have the largest average thickness. Furthermore it can be seen that the width of the distri-bution is similar, independent of the position with respect to the walls of the compression molding machine. Typically the non-isothermal history adds 0.04 to the FWHM of the lamellar thick-ness distribution. To compare the predicted average lamellar thickness distributions with the experimentally obtained ones, they are plotted in Fig. 13(b)for the different cooling rates. The agreement between predictions and experiments is good for both grades (only results for iPP-1 are shown). The most probable lamellar thickness, as well as the corresponding width of theFig. 10. Correction of the observed intensity for electron densityfluctuations and diffuse phase boundaries (left) and an interface distance distribution function obtained from a quasi-isothermal crystallization experiment on iPP-1. The Gaussians obtained from deconvolution indicate the cross-hatch distance distribution (line), the amorphous layer thickness distribution (dashed line), the lamellar thickness distribution (dots) and the long period distribution (dash-dotted).
Fig. 11. The lamellar thickness obtained from the interface distance distribution function (IDF) as a function of the lamellar thickness obtained from Bragg's law. Filled markers are from isothermal crystallization experiments and open markers are ob-tained from non-isothermal crystallization experiments.
distributionsfit the experimental data quite well, and differences of the average lcare within 5%.
6.4. Yield stress predictions
The lamellar thickness distributions obtained are used to predict the yield stress. Either the average lamellar thickness, or the lamellar thickness distribution can be used as an input for the relation between lamellar thickness and rate constant, see Eq.(31). When lamellae of all thicknesses contribute equally to the resis-tance against yielding there is actually no difference between the two procedures. To validate the yield stress predictions, experi-ments at room temperature and at 80C at a strain rate of 103s1 are carried out. At room temperature, both deformation mecha-nisms contribute to the yield stress whereas at 80 C only the contribution of the intra lamellar deformation process contributes to the yield stress. InFig. 14it is shown that quantitative agreement is found for both mold temperatures. The differences in yield stress that can be seen between the two grades are in quantitative agreement with the predictions.
Although the two iPP grades used in this study are relatively similar in terms of molecular weight and polydispersity some clear differences can be observed in the level of the yield stress. These differences arefirst of all reflected in the crystallization model. The nucleation density as a function of temperature and pressure is different for the two grades and can be expected to be unique for every material. The maximum growth rate of the alpha crystals is slightly different for the two materials and as a result of these
differences in the crystallization kinetics the range of crystallization temperatures in a non-isothermal cooling process is different. To relate crystallization kinetics and accompanying crystallization temperature to the formation of structural features like, in this case, the lamellar thickness, a material specific relation between these quantities had to be determined. In this work the inequalities were attributed to the differences in molecular chain architecture, and captured by varying the value for the surface free energy
s
e. The relation between lamellar thickness and yield stress contains the same parameters for both the materials, except for the rate con-stant which follows directly from the average lamellar thickness. 7. ConclusionThe 1D heat balance was successfully combined with a crystal-lization model capable of predicting the kinetics in non-isothermal pressure dependent quiescent conditions, and enabled us to predict the time-temperature history of two different iPP-grades cooled in a compression molding machine at different rates. Besides the nucleation density as a function of temperature and the maximum growth rate of the
a
crystals all parameters in the crystallization model were the same for the two grades used in this work. The temperature predictions were experimentally validated using in-situ temperature measurements. The amount of crystal volume as a function of time and temperature followed from the model framework as well, and was used in combination with the Lauritzen Hoffman equation to predict the lamellar thickness distributions formed during the different cooling histories. By only varying theFig. 12. Temperature measurements in isothermal crystallization experiments (left), and the relation between the lamellar thickness and the crystallization temperature (right). Markers represent experimental data and lines are bestfits. Filled markers are obtained from the isothermal experiments conducted on iPP-1 and iPP-2.
Fig. 13. Predictions of the lamellar thickness distribution as a function of the position with respect to the walls of the mold in an iPP-1 sheet, cooled in a cold press set at 20C (left).
surface free energy, which is known to be molecular weight dependent, experimentally obtained data from multiple authors could be described. Therefore it was chosen tofind a description of the lamellar thickness as a function of the crystallization temper-ature byfitting the surface free energy on data measured in-situ during an isothermal crystallization experiment. From this rela-tion lamellar thickness distriburela-tions were predicted that were experimentally validated using the interface distribution function. Good agreement was found for both the iPP grades, not only in terms of the average lamellar thickness, but also in terms of the FWHM of the distribution. This confirms that under the cooling conditions applied in this study the Lauritzen-Hoffman equation can be used, despite the absence of isothermal conditions in time and position. Finally, the lamellar thickness distributions were used together with a corrected empirical relation between lamellar thickness and rate constant reported in the work of van Erp et al., Eq.(36). This enabled us to predict the yield stress directly after processing at all loading conditions, i.e. strain rate and temperature. Predictions were made for a strain rate of 103and temperatures of 23C and 80C tensile tests were carried out for validation and the predicted yield stresses of both the iPP grades showed good agreement with the experimentally obtained data at all loading conditions. This work shows that making the connection between processing and mechanical properties is feasible. Extension toflow and multiple crystallographic structures is part of future work. Acknowledgments
The authors would like to thank the staff at beamline BM26 of the ESRF in France for the help during the experiments. Further-more, G. Portale is acknowledged for the valuable discussions about the data treatment. We thank NWO for granting beamtime to perform these experiments.
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